Lukas Müller

• In 2002, Korn and Wilmott introduced the worst-case scenario optimal portfolio approach. They extend a Black-Scholes type security market, to include the possibility of a crash. For the modeling of the possible stock price crash they use a Knightian uncertainty approach and thus make no probabilistic assumption on the crash size or the crash time distribution. Based on an indifference argument they determine the optimal portfolio process for an investor who wants to maximize the expected utility from final wealth. In this thesis, the worst-case scenario approach is extended in various directions to enable the consideration of stress scenarios, to include the possibility of asset defaults and to allow for parameter uncertainty. Insurance companies and banks regularly have to face stress tests performed by regulatory instances. In the first part we model their investment decision problem that includes stress scenarios. This leads to optimal portfolios that are already stress test prone by construction. The solution to this portfolio problem uses the newly introduced concept of minimum constant portfolio processes. In the second part we formulate an extended worst-case portfolio approach, where asset defaults can occur in addition to asset crashes. In our model, the strictly risk-averse investor does not know which asset is affected by the worst-case scenario. We solve this problem by introducing the so-called worst-case crash/default loss. In the third part we set up a continuous time portfolio optimization problem that includes the possibility of a crash scenario as well as parameter uncertainty. To do this, we combine the worst-case scenario approach with a model ambiguity approach that is also based on Knightian uncertainty. We solve this portfolio problem and consider two concrete examples with box uncertainty and ellipsoidal drift ambiguity.